In the Scissor Sisters’ song “I Don’t Feel Like Dancing,” one of the lyrics is “I don’t feel like dancin’ when the old Joanna plays.” Who is Joanna?
Cockney rhyming slang: Joanna = piano.
But that doesn’t rhyme? 
Sounds good. The song has a great piano accompaniment.
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Sending Joanna over to Cafe Society.
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It does if you pronounce it ‘pianna’ 
Cockney slang is this kooky thing that’s usually a couple times removed from the original word.
“Fancy a Butcher’s?” = “Fancy a look?” because hook rhymes with look, so it becomes “fancy a hook” and then eventually “fancy a butcher’s?”
Weirdos.
Don’t tell that to Cecil. (Skip to last line.)
Not quite. The rhyme was originally the full “butcher’s hook” which then got shortened to “butcher’s”. That’s probably what you meant anyway but other people might be confused.
But “joanna” is not a case like this, anyway. It simply rhymes with the usual Cockney pronunciation of “piano”, which is roughly “pee-ANN-a”.
In Australia, it’s “goanna”! See also large predatory lizard (Goanna - Wikipedia)